Programme

The schedule of the congress is here.

Talks

The titles and abstracts of the talks will be posted as soon as we get them.

Anna Miriam Benini (Pisa):

*Rigidity for non-recurrent exponential maps*

__An exponential map__

*f*(

*z*)=

*e*

^{z}+

*c*is called non-recurrent if the asymptotic value

*c*is not in the accumulation set of its own forward orbit. We will present the result that whenever two non-recurrent exponential maps satisfy some combinatorial equivalence, then they are conjugate by a quasiconformal map. If moreover

*c*has a bounded orbit then the conjugation can be made conformal.

François Berteloot (Toulouse):

*Bifurcation currents in holomorphic dynamics*

I will survey most of the results related to bifurcation currents for holomorphic families on the Riemann sphere and will discuss some generalization in higher dimension.

Filippo Bracci (Roma Tor Vergata):

*Fractional singularities of semigroups and evolution of multi-slits*

The theory of semigroups is pretty well established in one and several complex variables. In particular, zeros of the infinitesimal generator at a boundary point are essentially related to fixed points of the semigroup. Recently, with M. D. Contreras and S. Diaz-Madrigal, we defined a new type of boundary singularities (regular poles) for infinitesimal generators, which, essentially due to Julia's lemma, are the worst singularities that might appear. We related such singularities with the so-called "beta-numbers" appearing in the Brennan conjecture. Such machinery allows to study evolution of multi-slits radial domains and give a new insight on it. In this talk we will discuss such results and also some new results on what we call fractional singularities of semigroups.

Julie Deserti (Paris-Basel):

*Degree growth of birational maps of the plane*

The sequence of iterative degrees of a birational map of the plane is known either to be bounded or to have a linear, quadratic or exponential growth. We shall speak about the classification elements of in finite order with a bounded sequence of degrees. We shall then describe the coefficients of the linear and quadratic growth and relate them to geometrical properties of the map. If time allows we shall speak about the dynamics of a curious family of birational maps with linear growth.

Adam Epstein (Warwick):

*Geometric Limits in Holomorphic Dynamics*

We give a formal definition of the notion of geometric limit of conformal dynamical systems, and a sketch proof of a Structure Theorem for the geometric limits of algebraically convergent sequences of rational maps.

Núria Fagella (Barcelona):

*Baker domains of meromorphic maps and weakly repelling fixed points*

In this talk we show that if a meromorphic transcendental map has a multiply connected Baker domain, then it must also have at least one weakly repelling fixed point (i.e. repelling or with derivative equal to one). This was the last remaining case in the proof of the following result (which was proven by Shishikura for rational maps): If

*f*is a meromorphic transcendental map with a disconnected Julia set, then

*f*has a weakly repelling fixed point. The historical motivation of this theorem was its corollary, namely that the Julia set of Newton's method of every entire map is connected or, equivalently, all its Fatou components are simply connected. To prove this theorem we previously show the existence of absorbing regions in Baker domains, a question which has been open for some time.

Thomas Gauthier (Amiens):

*Strong bifurcation loci of maximal Hausdorff dimension*

In the moduli space of degree

*d*rational maps of the complex projective line, the bifurcation locus is the support of a closed positive (1,1)-current

*T*

_{bif}, which is called the bifurcation current. This current gives rise to a measure

*m*

_{bif}, which is the maximal self-intersection of

*T*

_{bif}, whose support is the seat of strong bifurcations. We will prove that the support of the measure

*m*

_{bif}has maximal Hausdorff dimension 2(2

*d*-2). As a consequence, the set of degree

*d*rational maps having 2

*d*-2 distinct neutral cycles is dense in a set of full Hausdorff dimension.

Pavel Gumenyuk (Roma Tor Vergata):

*Loewner equations, evolution families and their boundary fixed points*

Loewner Theory proved to be a powerful tool in Complex Analysis. One might mention its crucial role in the proof of the Bieberbach Conjecture on sharp coefficient estimates for univalent functions and the recently discovered by O. Schramm stochastic version of Loewner Evolution playing now an important role in Statistical Physics. This talk is devoted to some recent developments in the (deterministic) Loewner Theory in the unit disk connected to a new approach proposed in 2008 by F. Bracci, M.D. Contreras and S. Díaz-Madrigal. According to this approach, one of the cornerstones of Loewner Theory is the notion of an evolution family, which can be regarded as a non-autonomous analogue for one-parameter semigroups of holomorphic self-maps. Among others we will present a new result on the boundary behaviour of evolution families relating regular boundary fixed points with the regular boundary null-points of the r.h.s. in the generalized Loewner ODE (joint work with F. Bracci, M.D. Contreras and S. Díaz-Madrigal).

John H. Hubbard (Marseille-Cornell):

*Parabolic blowups and geometric limits of dynamical systems*

The filled in Julia set of a polynomial does not depend continuously on the polynomial. A natural question to ask is: what is the closure of the space of filled in dynamical systems? I will answer this question in the case of quadratic polynomials, including a computation of the cohomology of the space. It is a lot more complicated than one might imagine. I will draw parallels with geometric limits of Kleinian groups.

David Marin Pérez (Barcelona):

*Topological classification of singular germs of foliations in the plane*

We review some recent topological facts, obtained in collaboration with Jean-François Mattei, about singular holomorphic foliations in the plane which allows us to introduce a new topological invariant of germs called monodromy. Our main result concerns the topological classification of germs of foliations satisfying weak generic condition on each equisingularity class and reads roughly as follows: Two such germs are topologically conjugated if and only if there is a special conjugation between their monodromies. As a corollary we obtain a proof of the Cerveu-Sad conjecture about the topological invariance of the projective holonomy representations.

Carsten Lunde Petersen (Roskilde):

*Double parabolic implosion in the quadratic family*

In a recent paper Buff and Epstein consider parabolic renormalization of the quadratic rational maps admitting a parabolic fixed point with multiplier 1 and multiplicity 2. Such maps are conveniently parametrized as

*f*

_{B}(

*z*) =

*z*+ 1 +

*B*/

*z*, with

*B*not zero. A parabolic renormalization of

*f*

_{B}is a Lavours horn map with parabolic multiplier some root of unity. Buff and Epstein studied the sets of parameters

*A*

_{p/q}for which say the upper horn map has multiplier exp(

*i*2

*π*

*p*/

*q*). They show, using rather elaborate techniques that each set

*A*

_{p/q}is an infinite and discrete set which accumulates only infinity. In the talk I will discuss another approach to these results, which shows that the sets

*A*

_{p/q}are asymptotically periodic at infinity with period 1/

*i*2

*π*and that a fundamental domain contains

*q*points from

*A*

_{p/q}. In particular I shall outline a proof that the points of

*A*

_{0}is a sequence {

*B*

_{n}}

_{n}with the asymptotic behavior

*B*

_{n}= 1/4 - 1/

*i*4

*π*+

*n*/

*i*2

*π*+

*o*(1). A similar formula has also recently been reported by Reinhard Schäfke and Augustin Fruchard.

Matteo Ruggiero (Paris):

*Contracting rate of iterates of superattracting germs in dimension 2*

Take a dominant superattracting holomorphic germ

*f*in dimension 2. We are interested in the growth of the local contracting rate

*c*(

*f*

^{n}) as

*n*tends to infinity. Using a detailed analysis of the dynamics induced on the valuative tree, we show that the sequence

*c*(

*f*

^{n}) satisfies an integral recursion formula (joint work with William Gignac).

David Sauzin (Pisa):

*On the obtention of the Écalle-Voronin invariants via resurgence theory*

For a holomorphic germ of the complex plane with a simple parabolic fixed point at the origin, we discuss the resurgent approach to the construction of the attracting and repelling Fatou coordinates and the description of the horn map which classifies such germs up to analytic conjugacy. The method relies on the use of the Borel-Laplace summation for the “formal iterator”, which is the asymptotic expansion common to both Fatou coordinates (joint work with Artem Dudko).

Dierk Schleicher (Bremen):

*News about Newton: global dynamics and efficient root finding*

Newton's method as a root finder is locally very fast, but the global dynamics seemed difficult to understand, even for complex polynomials in one variable. In particular, there may be open sets of starting points that converge to no roots (but to attracting cycles of higher periods). We describe how to turn Newton's method into an efficient root finder that, in the expected case, needs no more than

*O*(

*d*

^{2}log

^{5}

*d*+

*d*log| log

*ε*|) iterations to find all roots of a degree

*d*polynomial with precision

*ε*. We also answer a question by Smale to classify all polynomials for which there are attracting cycles of higher periods (joint with Mikulich).

Nessim Sibony (Paris):

*Dynamics of regular polynomial automorphisms of C*

^{k}

For the abstract click here.

Vladlen Timorin (Moscow):

*Topological cubic polynomials*

As was noted by Thurston, the topological dynamics of polynomials with locally connected Julia sets can be described in terms of invariant laminations, i.e. invariant equivalence relations on the circle satisfying certain natural properties. Invariant laminations provide a slightly wider class of topological dynamical systems (called topological polynomials) than those coming from polynomials with locally connected Julia sets. We study topological cubic polynomials and the structure of their parameter space. In particular, we will discuss the combinatorial main cubioid. It consists of laminations corresponding to polynomials on the boundary of the principal hyperbolic component. (Based on a joint project with Alexander Blokh, Lex Oversteegen and Ross Ptacek)

Sergei Yakovenko (Rehovot):

*Intersection multiplicity growth in local dynamical systems*

*If*

*X*and

*Y*are two submanifolds of complimentary directions in a smooth manifold

*M*, and

*F*a diffeomorphism of

*M*into itself, then one can drag

*X*by iterates of

*F*and look at the number of intersections with

*Y*. Generically the number grows exponentially with the number of iterations, but (as was discovered by Arnold in 1990) if

*F*is a germ of a diffeomorphism at a fixed point, the multiplicity of intersection remains bounded. The results are directly related to the Artin-Mazur theorem on the growth of periodic points and Shub-Sullivan theorem on the boundedness of index of the iterates. Arnold's proof rests upon the Skolem theorem from the logic/number theory. We show that the above boundedness holds for any holomorphic action of any commutative subgroup of germs of self-maps at a fixed point. Moreover, the phenomenon can be explained by simple Noetherianity-type arguments for the rings of quasipolynomials. (Joint work with Anna L. Seigal, Cambridge U.)

Dmitri Zaitsev (Dublin):

*Dynamics of multi-resonant biholomorphisms*

Our goal is to study the dynamics of holomorphic diffeomorphisms in C

^{n}such that the resonances among the first

*r*eigenvalues of the differential are generated by a finite number of linearly independent multi-indices (and more resonances are allowed for other eigenvalues). We give sharp conditions for the existence of basins of attraction where a Fatou coordinate can be defined. (This is a joint work with Filippo Bracci and Jasmin Raissy)

Anna Zdunik (Warsaw):

*Fine inducing in holomorphic dynamics*

I will describe an inducing scheme, which turned out to be very useful in the study of dynamics of holomorphic maps. A series of results, in particular stochastic properties of natural invariant measures, and some ridigity theorems, can be shown, using this common scheme.